# Modeling bidirectional communication across a coax?

I was reading about the history of cable internet and I'm interested in a simple model of full-duplex communications across a single coaxial cable. My first attempt is simply a transmission line with a voltage source at either end.

Either party can send a signal using their voltage source and receive a signal by listening to the voltage across the coax line (voltage detector not shown). This model cannot be right, because what if the left-hand party decides to send a constant 0 signal while the right-hand party wants to send a constant sine wave simultaneously? The left-hand party necessarily shorts his side of the coax which will ruin any ability to detect the sine wave.

Here is my second model where I've only drawn the right-hand party.

They send a signal by driving Vout and detect the signal by reading Vin. The two resistors have very high resistance values. How accurate is my model? Would something like this basically work for full-duplex communication?

This arrangement is missing the termination which is typically seen in a transmission line circuit -- when terminated, both sides drive their respective ends of the transmission line with voltage sources through some non-zero impedance; sensing the voltage at the ends of the line yields a superposition of the signals from both ends.

As a bonus, there are also no reflections when the line is properly terminated.

Here's an example of series termination. Note the impedance looking into each endpoint is 50 ohms, since it's a 50-ohm resistor in series with a voltage source (which presents zero impedance itself). This is actually somewhat similar to the second schematic in your question, although it uses one less resistor.

At each red node, the voltage observed is half the voltage of the closest source, plus a delayed version (due to propagation delay) of half the voltage of the distant source. The one-half factor is a result of voltage division between the 50 ohm impedance of the local termination and the 50 ohm impedance of the transmission line + distant termination. This quick analysis holds because the termination is well-matched; if there were mismatches, then the mismatched impedance would need to be transformed across the transmission line, for example with the aid of a Smith Chart.

A similar discussion can be made for parallel termination with current sources - when unterminated, the two sources try to inject different currents into the line, and end up leading to a conflict when solving for currents in the circuit. Parallel termination across those current sources will make the math work and make the circuit likewise work.

• To arrive at the one-half superposition, you are simply doing basic circuit theory KCL / KVL / Ohms law, right? And then the delay comes in due to transmission line theory.
– Mark
Jul 10, 2021 at 0:28
• @Mark that's correct. It holds for the case of no reflections because the termination matches the Z0 of the transmission line. If the termination were not matched to the line at one or both ends, a more complex frequency-dependent impedance transformation will be necessary. Jul 10, 2021 at 0:30

You need hybrids at each end - something that can separate the waves travelling in each direction, something like: (V1 and V2 are the signals from each end)

simulate this circuit – Schematic created using CircuitLab

There are ways of doing it with transformers that telephones used to use, and also in RF there are a whole raft of hybrids / directional couplers.

In telephones you usually don't want perfect separation, sidetone is beneficial.